I'm going to be a little difficult here for a second..
Does your gym have an area, where you'd feel comfortable that has mats and some dumbbells? Most gyms I've seen have an open area that isn't part of the free weights that has this. If not an empty class room that has a variety of dumbbells and a mat would be suitable too. If that fails you, then hopefully they have a cable machine.
For the arms (and strictly arms) and abs, machines generally are very poor. The arms are composed of smaller muscles and I always found the machines that you did find to be a bit uncomfortable and unsuited for women in general (larger weight increases).
I read a few years ago that there was a study done on a variety of ab exercises including machines and plain old crunches was the winner. Planks (regular and side) are also a good complimentary exercise. If there is no mat area to use at your gym, then I'd recommend doing crunches and planks at home.
One of the best overall arm and upper body exercises is the pushups which you can do on your knees or on an incline (such as steps or an exercise step) if you aren't able to do a full pushup. I do mine on my steps at home but the gym I went to used to have some various sized steps I used as well.
Other than that, there is the good old bicep curl which can be done with dumbbells and tricep kickbacks.
(this shows sitting down but I'd recommend standing up)
So if there isn't a mat area or ability to grab a couple dumbbells, then I'd look for a cable machine. Cable machines are multifunctional and usually have a cable hook up above you and one near the ground. It may also be adjustable to where you put the cable.
The bicep curl and a tricep pushdown can be done on the cable machine. Also realize that if you are standing up with your back against nothing, you are most likely working the stabilizer muscles in your abs.
Also, one of the best exercises overall for the upper body is the pullup which most women unless they are near goal would not be able to do. Luckily, the cable machines usually have a seated bench too or your gym may have something called an assisted pullup machine. They work your arm muscles and your back muscles.
I know that isn't exactly the answer you were looking for but based on the muscle groups you specified, that is what I'd recommend.